Sounds like you have some plumbing repairs ahead, luckily the plumbing system is pretty basic.
When hooked up to city water, plumbing fixtures are supplied by the hose pressure, the pump is not used. You do need to use a pressure regulator to avoid damage to your plumbing system - available at any RV parts store.
When using water from the tank, the pump pressurizes the system. There is a swithch - probably in the kitchen to turn the pump on. Leave it on when camping, the pump is an ondemand unit and will only pump when pressure is needed.
If the pump runs continuously there is either a leak - pipe or tap, or the pump needs to be rebuilt/replaced. The pump should build pressure then automatically shut off until you open a tap. You have to open all taps to purge the system of air after winterizing or you will get spurts and bubbles from your taps and the pump may run on trying to pressurize air in the system.
If there is water comming out where it shouldn't you probably have a split pipe from improper winterization.
My 77 had a bad pump and several split pipes. I replaced all the plumbing pipe with Pex, new pump from Camping world and new taps/ fixtures. This way I know I have a system I can rely on.
The waste valve may be stuck due to waste hardning on the valve. Try adding water to the tank (don't fill it up) then tow around and try to break up the blockage. You may have to replace the waste gatevalve, but parts are available.
1977 Safari Land Yacht
2005 Toyota Tundra SR5
2010 Ford Flex Ecoboost